Decreasing Energy Functions and Lengths of Transients for Some Cellular Automata

Th e work of Ghiglia, Masti n, an d Rom ero on a "p hase-unwrapping" algorith m gives rise to the following operation : for any undirect ed graph with a rbitra ry int eger values attached to the verti ces, simultaneous upd at es are perform ed on these values, with t he value of a verte x being cha nged by one in th e direction of th e average of t he values of the adjace nt vert ices. (When t he average equals t he value of a vertex, th e val ue of th e ver tex is incremented by one, unless all the neighbors have th e same value, in which case no change is made.] Ea rlier work of Odlyzko and Rand all showed that iterat ing t his operation always leads to a cycle of length one or two, bu t did not give a bound on how ma ny iterations might be needed to reach such a cycle. T his paper int rodu ces a new "energy funct ion" which does yield a bound for the t rans ient. A novel feature of this energy is that it contains not only linea r an d bilinea r ter ms, as is common, but also terms involving th e minimum function. 1. In t r o d u c ti on Let G b e an undirected graph with ve rt ices lab elled 1,. .. , n, and su pp ose t hat for each i, an integer Xi (O) is as sign ed at time 0 to vertex i. A se ries of sy nch ro nous updates is p er formed on t hese val ue s with t he rule that if Xi (t) is the valu e o f vertex i at t ime t, t hen { Xi(t)-1 if Silt) < 0, Xi(t + 1) = Xi(t) if Xj(t) = Xi(t) for all j E J i , Xi(t) + 1 other wise, "Sup ported partially by AT&T Bell Labor a tori es and Fondo Naciona l de Ciencias de Chile 86·8 7, Sa nt iago, Chile.